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ABSTRACT
This article deals with improved estimation of a Weibull (a) shape parameter, (b) scale parameter, and (c) quantiles in a decision-theoretic setup. Though several convenient types of estimators have been proposed in the literature, we rely only on the maximum likelihood estimation of a parameter since it is based on the sufficient statistics (and hence there is no loss of information). However, the MLEs of the parameters just described do not have closed expressions, and hence studying their exact sampling properties analytically is impossible. To overcome this difficulty we follow the approach of second-order risk of estimators under the squared error loss function and study their second-order optimality. Among the interesting results that we have obtained, it has been shown that (a) the MLE of the shape parameter is always second-order inadmissible (and hence an improved estimator has been proposed); (b) the MLE of the scale parameter is always second-order admissible; and (c) the MLE of the p-th quantile is second-order inadmissible when p is either close to 0 or close to 1. Further, simulation results have been provided to show the extent of improvement over the MLE when second-order improved estimators are found.
Acknowledgment
The authors thank three anonymous referees for many helpful comments and suggestions, which helped immensely in the presentation of this work.
Funding
For the second author, this research is funded by the Foundation for Science and Technology Development of Ton Duc Thang University (FOSTECT), website: http://fostect.edu.vn, under grant FOSTECT.2015.BR.20.